FDTD改进算法及其理想导体边界实现

发布时间：2018-01-04 11:01

本文关键词：FDTD改进算法及其理想导体边界实现　出处：《西南交通大学》2014年博士论文　论文类型：学位论文

【摘要】：计算电磁学是电磁场理论、数学和计算机技术相结合的产物。计算电磁学正向着高精度、高效能、高速度的目标快速发展,很多以前无法解决的疑难电磁场问题找到了很好的解决方法,得到了精确的计算结果。越来越多的实际工程电磁场疑难问题摆在人们面前,这更促进了计算电磁学向更高水平发展。基于麦克斯韦方程组的解析方法和数值方法求解是计算电磁学的主要任务。通常,只有经典的电磁场问题才有解析解,数值计算逐渐成为解决复杂电磁场问题的主要手段,有时甚至是唯一手段。FDTD方法是典型的时域全波分析方法,应用范围非常广泛,是近些年最受关注、发展最迅速的数值方法之一。麦克斯韦方程是描述电磁现象的基本方程,FDTD方法从其旋度方程,即麦克斯韦微分形式出发,对时间和空间的一阶偏导数采取中心差分近似,直接转换为显式差分运算,这样可以在时间和空间上对连续电磁场数据实现抽样离散。FDTD方法能够描述时域电磁场的传播特性,只要给出问题的初始条件和边界条件,即可应用FDTD方法迭代递推得到各个时间步和空间步的电磁场分布。随着计算机硬件条件的发展,与FDTD方法相关的创新研究不断涌现,FDTD方法将会赢得越来越多的计算电磁学领域专家的关注和青睐。然而,FDTD方法也有其自身不可忽视的不足：一方面,采用差分法对麦克斯韦方程近似求解时,会在计算网格中引起数值色散,这种关系随数值波模的传播方向以及离散化程度不同而改变。因此,FDTD方法受限于数值色散条件,一般空间步长不大于波长的十分之一,当数值模拟对象的电尺寸较大时,将导致所需内存剧增；另一方面,FDTD方法的时间步长和空间步长不独立,Courant-Friedrich-Levy (CFL)稳定条件限制了时间离散,使得时间步长必须随空间步长变化而变化,如果空间步长变小,则需要增加时间迭代步数以实现收敛。因此,对于目前普通的PC机而言,FDTD方法在计算效率方面存在着不足。围绕着电磁场数值方法的计算效率问题,近年来出现了FDTD的多种改进算法：针对内存问题的R-FDTD方法,针对计算时间步长问题的ADI-FDTD和LOD-FDTD方法都是较热门的数值计算方法。当然,在实际应用中,会发现这些算法都或多或少存在一些问题,对这些FDTD方法的改进算法作进一步的研究,以节约内存使用量和计算时间,提高计算准确程度为目的,实现更加高效、更加精确的数值计算,具有理论和应用意义。论文首先对课题的研究背景及意义进行了阐述,包括计算电磁学进展、发展时域数值计算的必要性以及时域有限差分方法的介绍。本文涉及到的FDTD改进算法包括R-FDTD方法、ADI-FDTD方法和LOD-FDTD方法。它们能够很好的解决内存使用量大,计算时间长的问题。介绍了几种FDTD改进算法的研究现状。最后说明了本文所做的主要工作。针对具有对称结构的计算模型,从理论上分析了采用PEC边界和PMC边界截断的对称边界条件,提出了对称截断FDTD方法,利用该方法能够确定截断边界以外场分量的值,以实现截断边界处的FDTD方法计算。数值计算验证了对称截断FDTD方法的正确性和可行性。论述了R-FDTD方法中对暂存场分量边值进行补充计算的必要性。论证了三维R-FDTD求解感应电荷密度处理导体问题的方法与传统FDTD方法等价。提出了在内存使用量和计算时间上都具有明显优势的周期对称结构R-FDTD方法,该方法结合了R-FDTD方法和第二章对称截断方法的优点,将计算区域的内存使用量最多降为FDTD算法的1/6。由于每个时间步迭代计算的复杂度降低,需要计算的网格数明显减少,总的迭代计算时间也大幅缩减。数值计算验证了该方法的正确性和有效性。为了准确求解ADI-FDTD方法实现PEC边界和PMC边界的待求场分量系数,通过在获得该系数前应用理想导体边界条件,推导出了相应的修正系数。计算了单个金属立方体和对称的两个金属立方体的双站RCS。结果表明,理想导体边界作为理想导体表面,采用修正系数的计算结果与FDTD方法计算结果更为吻合；理想导体边界作为截断计算空间对称面,采用修正系数的计算结果与ADI-FDTD方法计算结果相同,与理论推导结论一致。证明了LOD-FDTD方法实现PEC边界和PMC边界时的待求场分量系数与传统的LOD-FDTD方法系数不同。通过在获得该系数前应用理想导体边界条件,得到对应的修正系数。针对将理想导体边界条件作为理想导体表面和截断计算空间对称面的不同情况,讨论了修正系数与传统LOD-FDTD系数的区别。具有统一的表达式的修正系数LOD-FDTD方法计算理想导体表面较传统LOD-FDTD方法误差更小,并对其进行了数值验证。
[Abstract]:Computational electromagnetics is the product of electromagnetic theory, mathematics and computer technology. The combination of computational electromagnetics towards high precision, high efficiency, rapid development of high speed target, many can not solve the difficult problem of electromagnetic field before to find a good solution, obtain the calculation precision. The actual engineering electromagnetic problems more and more in front of people, it promotes the computational electromagnetics to a higher level of development. To solve the analytical and numerical methods based on Maxwell's equations is the main task of computational electromagnetics. Usually, only by classical electromagnetic field problems has analytic solution, numerical calculation has become the main means to solve complex electromagnetic problems, sometimes even is the only.FDTD method is a typical time-domain full wave analysis method, application range is very wide, is the most popular in recent years, the rapid development of numerical One of the methods. The Maxwell equation is the basic equation describing the electromagnetic phenomena, the FDTD method from the view that the Maxwell curl equations, the differential form of time and space to take the first derivative central difference approximation is directly converted into explicit difference operation, the propagation characteristics of this can in time and space of continuous electromagnetic data sampling discrete.FDTD method can describe the time-domain electromagnetic field, as long as the problem is given the initial conditions and boundary conditions, the distribution of electromagnetic field can be used FDTD iterative recursive method to get each time step and spatial step. With the development of computer hardware, and FDTD related innovative research methods are constantly emerging, the FDTD method will win the attention and favor more and more experts in the field of computational electromagnetics. However, the FDTD method has its own shortcomings can not be ignored: on the one hand, the difference of Mike The approximate solution of Maxwell equations, will cause the numerical dispersion in computational grid, the relationship with the numerical wave propagation direction and discrete degree of different changes. Therefore, the FDTD method is limited to numerical dispersion conditions, general space step size is not greater than the wavelength of the 1/10, when the electrical large size when the simulation object, will cause to increase memory; on the other hand, the time step and spatial step FDTD method is not independent, Courant-Friedrich-Levy (CFL) stable conditions of discrete time, the time step must change with the change of space, if space is smaller, you will need to increase the time step iteration convergence hundreds. Therefore, for the PC machine at present in general, the FDTD method in computing efficiency shortcomings. Around the computation efficiency of the numerical method, in recent years there has been a variety of improved FDTD algorithm Method: R-FDTD method for memory problems, according to the ADI-FDTD and LOD-FDTD method to calculate the time step problem is numerical calculation of popular methods. Of course, in practice, will find that these algorithms are more or less there are some problems and improvement of these FDTD algorithm for further study, in order to save the use amount and calculation time memory, improve the calculation accuracy for the purpose of achieving more efficient and more accurate numerical calculation, which has theoretical and practical significance. Firstly, the research background and significance are expounded, including the development of computational electromagnetics, computational development and the necessity of time-domain numerical finite-difference time-domain method is introduced. This article relates to the the improved FDTD algorithm including R-FDTD method, ADI-FDTD method and LOD-FDTD method. It can solve the memory usage and long computing time. The research status of several improved FDTD algorithm. Finally the main work done in this paper. According to the calculation model with symmetrical structure, from the theoretical analysis of the symmetric boundary conditions of the PEC and PMC boundary truncation, the symmetric truncated FDTD method, using this method can determine the truncation boundary to field values. In order to implement the FDTD method the truncation boundary calculation. The numerical results verify the correctness and feasibility of truncated FDTD method. The R-FDTD method of temporary field boundary value necessary to supplement calculation. Demonstrates the equivalence method of 3D R-FDTD for induction treatment and charge density of the conductor. The traditional FDTD method put forward cycle symmetrical R-FDTD method has obvious advantages in memory usage and computation time, this method combines the R-FDTD method and the second chapter symmetric truncation method. That reduces the complexity of the computational region of the memory usage of the most reduced FDTD algorithm 1/6. calculation because each time step iteration, the number of grid computing needs significantly reduced, the total computing time is significantly reduced. The numerical results verify the correctness and validity of the method. In order to accurately solve the implementation method of PEC ADI-FDTD and PMC boundary unknown field coefficients, through the application of ideal conductor boundary conditions in the coefficients are derived, corresponding correction factor. The calculation of the two metal cube single metal cube and symmetrical bistatic RCS. results show that the ideal conductor boundary as ideal conductor surface, calculation results using the results correction the coefficient and FDTD method is more consistent; as the perfect conductor boundary to truncate the computational space plane of symmetry, the calculation results of correction coefficient method and ADI-FDTD calculation results At the same time, consistent with the theory. The results show that the LOD-FDTD method to achieve PEC and PMC boundary when the undetermined coefficient method of LOD-FDTD coefficients and the traditional field. Through the application of ideal conductor boundary conditions in the coefficient, the correction coefficient has been obtained. The corresponding ideal conductor boundary conditions as the ideal conductor surface and truncation calculation the spatial symmetry of different situations, different with the traditional correction coefficient LOD-FDTD coefficient is discussed. The modified LOD-FDTD method has uniform coefficient expression of the calculation error of ideal conductor surface than the traditional LOD-FDTD method, and carries on the numerical simulation.

【学位授予单位】：西南交通大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TM15

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